Curl-free Wavelets Research Articles

Biorthogonal wavelets and tight framelets from smoothed pseudo splines

In order to get divergence free and curl free wavelets, one introduced the smoothed pseudo spline by using the convolution method. The smoothed pseudo splines can be considered as an extension of pseudo splines. In this paper, we first show that the shifts of a smoothed pseudo spline are linearly independent. The linear independence of the shifts of a pseudo spline is a necessary and sufficient condition for the construction of the biorthogonal wavelet system. Based on this result, we generalize the results of Riesz wavelets and derive biorthogonal wavelets from smoothed pseudo splines. Furthermore, by applying the unitary extension principle, we construct tight frame systems associated with smoothed pseudo splines with desired approximation order.

A class of generalized pseudo-splines

In this paper, a class of refinable functions is given by smoothening pseudo-splines in order to get divergence free and curl free wavelets. The regularity and stability of them are discussed. Based on that, the corresponding Riesz wavelets are constructed.

Three-Dimensional Biorthogonal Divergence-Free and Curl-Free Wavelets with Free-Slip Boundary

This paper deals with the construction of divergence-free and curl-free wavelets on the unit cube, which satisfies the free-slip boundary conditions. First, interval wavelets adapted to our construction are introduced. Then, we provide the biorthogonal divergence-free and curl-free wavelets with free-slip boundary and simple structure, based on the characterization of corresponding spaces. Moreover, the bases are also stable.

Anisotropic curl-free wavelet bases on the unit cube

This paper deals with the construction of anisotropic curl-free wavelets on the cube [0, 1]3, which satisfies the specific boundary conditions. First, one constructs curl-free wavelets on the unit cube based on one dimensional wavelets on the interval [0, 1] with some boundary conditions. Then, the stability of the corresponding wavelets in curl-free space and the characterization of Sobolev spaces are studied. Finally, one gives a Helmholtz decomposition and the representation of curl and div operators in wavelet coordinates.

Anisotropic curl-free wavelets with boundary conditions

This paper deals with the construction of anisotropic curl-free wavelets that satisfy the tangent boundary conditions on bounded domains. Based on some assumptions, we first obtain the desired curl-free Riesz wavelet bases through the orthogonal decomposition of vector-valued L 2 . Next, the characterization of Sobolev spaces is studied. Finally, we give the concrete construction of wavelets satisfying the initial assumptions. MSC: 42C20

Interpolatory curl-free wavelets on bounded domains and characterization of Besov spaces

Based on interpolatory Hermite splines on rectangular domains, the interpolatory curl-free wavelets and its duals are first constructed. Then we use it to characterize a class of vector-valued Besov spaces. Finally, the stability of wavelets that we constructed are studied.MR(2000) Subject Classification: 42C15; 42C40.

Orthogonal Helmholtz decomposition in arbitrary dimension using divergence-free and curl-free wavelets

We present tensor-product divergence-free and curl-free wavelets, and define associated projectors. These projectors enable the construction of an iterative algorithm to compute the Helmholtz decomposition of any vector field, in wavelet domain. This decomposition is localized in space, in contrast to the Helmholtz decomposition calculated by Fourier transform. Then we prove the convergence of the algorithm in dimension two for any kind of wavelets, and in larger dimension for the particular case of Shannon wavelets. We also present a modification of the algorithm by using quasi-isotropic divergence-free and curl-free wavelets. Finally, numerical tests show the validity of this approach for a large class of wavelets.

INTERPOLATORY CURL-FREE WAVELETS AND APPLICATIONS

Divergence-free and curl-free vector wavelets have many applications for the analysis and numerical simulation of incompressible flows and certain electromagnetic phenomena. Because special domains are required in those applications, Bittner and Urban constructed interpolating divergence-free multi-wavelets based on cubic Hermite splines in 2005. In this paper, we construct interpolating curl-free multi-wavelets and give two wavelet estimates for a class of vector Besov spaces.

Décomposition de Helmholtz par ondelettes : convergence d'un algorithme itératif

In what follows, we present tensor-product divergence-free and curl-free wavelets, and we define associated projectors. These projectors permit the construction of an iterative algorithm for the computation of the Helmholtz decomposition in terms of wavelets. This Helmholtz decomposition is localized in space, in contrast to a Helmholtz decomposition calculated by the Fourier transform. Finally, we show the convergence of the algorithm in 2 and 3 dimensions for the particular case of Shannon wavelets.

Divergence-free and curl-free wavelets in two dimensions and three dimensions: application to turbulent flows

We investigate the use of compactly supported divergence-free wavelets for the representation of solutions of the Navier–Stokes equations. After reviewing the theoretical construction of divergence-free wavelet vectors, we present in detail the bases and corresponding fast algorithms for two and three-dimensional incompressible flows. We also propose a new method for practically computing the wavelet Helmholtz decomposition of any (even compressible) flow; this decomposition, which allows the incompressible part of the flow to be separated from its orthogonal complement (the gradient component of the flow) is the key point for developing divergence-free wavelet schemes for Navier–Stokes equations. Finally, numerical tests validating our approach are presented.